Extrapolated phases

Fig. 6.41 Calculated constant xN (=11.0) phase diagram for a blend containing equal amounts of two homopolymers and a symmetric diblock, all with equal chain length (Janert and Schick 1997a). The region of three-phase coexistence between ordered lamellar phases is shaded. Extrapolated phase boundaries are shown with dashes.

From the various possible closures, the mean spherical approximation (MSA) [189] has found particularly wide attention in phase equilibrium calculations of ionic fluids. The Percus-Yevick (PY) closure is unsatisfactory for long-range potentials [173, 187, 190]. The hypemetted chain approximation (HNC), widely used in electrolyte thermodynamics [168, 173], leads to an increasing instability of the numerical algorithm as the phase boundary is approached [191]. There seems to be no decisive relation between the location of this numerical instability and phase transition lines [192-194]. Attempts were made to extrapolate phase transition lines from results far away, where the HNC is soluble [81, 194]. 

Figure 11.15. Neglecting the existence of a high-melting line compound (AS) in the phase equilibria calculations (left) can result in a largely erroneous liquidus or solidus in an extrapolated phase diagram (right).

FIGURE 11.9 Phase diagram for conformationally symmetric diblock-copolymer melts showing regions of stability for the disordered (D), lamellar (L), gyroid (G), hexagonal (H), and cubic (C) phases. Dashed lines denote extrapolated phase boundaries and the dot marks the critical point. (Adapted from Matsen, M.W. and Schick, M., Phys. Rev. Lett., 72, 2660, 1994. With permission from the American Physical Society.)... 

Figure 13.13 Mean-field phase diagram tor confoimationally symmetric diblock copolymer melts. Phases are labeled L, lamellar H, hexagonal cylinders Qia3d, bicontinuous Ia3d cubic Qimsm, bcc spheres CPS, close-packed spheres and DIS, disordered region. The dashed lines denote extrapolated phase boundaries, and the dot denotes the mean-field critical ODT.

Large errors in the low-pressure points often have little effect on phase-equilibrium calculations e.g., when the pressure is a few millitorr, it usually does not matter if we are off by 100 or even 1000%. By contrast, the high-pressure end should be reliable large errors should be avoided when the data are extrapolated beyond the critical temperature. 

In the annealed cylinder the presence of own compressive stress has been observed, which was disclosed as lack of indications changes in the initial stretching phase, and their value may be evaluated on the basis of extrapolating the graphs to the coordinate axis. 

The separation of the solid phase does not occur readily with some liquid mixtures and supercooling is observed. Instead of an arrest in the cooling curve at /, the cooling continues along a continuation of c/ and then rises suddenly to meet the line f g which it subsequently follows (Fig. 1,13, 1, iii). The correct freezing point may be obtained by extrapolation of the two parts of the curve (as shown by the dotted line). To avoid supercooling, a few small crystals of the substance which should separate may be added (the process is called seeding ) these act as nuclei for crystallisation. 

This relationship is sketched in Fig. 4.7a, which emphasizes that P, must vary linearly with 6 and that P, ° must be available, at least by extrapolation. The heat of fusion is an example of a property of the crystalline phase that can be used this way. It could be difficult to show that the value of AH is constant per unit mass at all percentages of crystallinity and to obtain a value for AHj° for a crystal free from defects. Therefore, while conceptually simple, the actual utilization of Eq. (4.37) in precise work may not be easy. 

This has the advantage that the expressions for the adsotbed-phase concentration ate simple and expHcit, and, as in the Langmuir expression, the effect of competition between sorbates is accounted for. However, the expression does not reduce to Henry s law in the low concentration limit and therefore violates the requirements of thermodynamic consistency. Whereas it may be useful as a basis for the correlation of experimental data, it should be treated with caution and should not be used as a basis for extrapolation beyond the experimental range. 

Density. Although the polymer unit cell dimensions imply a calculated density of 1.33 g/cm at 20°C, and extrapolation of melt density data indicates a density of 1.13 g/cm at 20°C for the amorphous phase, the density actually measured is 1.15—1.26 g/cm, which indicates the presence of numerous voids in the stmcture. 

When liquid-phase resistance is important, particular care should be taken in employing any given set of experimental data to ensure that the equilibrium data used conform with those employed by the original author in calculating values of fci or Hi. Extrapolation to widely different couceutratiou ranges or operating conditions should... 

Extrapolation of KgO data for absorption and stripping to conditions other than those for which the origin measurements were made can be extremely risky, especially in systems involving chemical reactions in the liquid phase. One therefore would be wise to restrict the use of overall volumetric mass-transfer-coefficient data to conditions not too far removed from those employed in the actual tests. The most reh-able data for this purpose would be those obtained from an operating commercial unit of similar design. 

The traditional design method normally makes use of overall values even when resistance to transfer lies predominantly in the liquid phase. For example, the COg-NaOH system most commonly used for comparing the Kg< values of various tower packings is a liqiiid-phase-controlled system. When the liqiiid phase is controlling, extrapolation to different concentration ranges or operating conditions is not recommended since changes in the reaction mechanism can cause /cl to vary unexpectedly and the overall values do not explicitly show such effects. 

The presence of errors within the underlying database fudher degrades the accuracy and precision of the parameter e.stimate. If the database contains bias, this will translate into bias in the parameter estimates. In the flash example referenced above, including reasonable database uncertainty in the phase equilibria increases me 95 percent confidence interval to 14. As the database uncertainty increases, the uncertainty in the resultant parameter estimate increases as shown by the trend line represented in Fig. 30-24. Failure to account for the database uncertainty results in poor extrapolations to other operating conditions. 

Aside from the fundamentals, the principal compromise to the accuracy of extrapolations and interpolations is the interaction of the model parameters with the database parameters (e.g., tray efficiency and phase eqiiilibria). Compromises in the model development due to the uncertainties in the data base will manifest themselves when the model is used to describe other operating conditions. A model with these interactions may describe the operating conditions upon which it is based but be of little value at operating conditions or equipment constraints different from the foundation. Therefore, it is good practice to test any model predictions against measurements at other operating conditions. 

Corresponding to this, the reactance from Figure 28.19(b) can be determined by extrapolation. We have assumed it to be 0.06 0/1000 m per phase. 

Total Interstitial Volume, value extrapolated from the retention volumes of ions of different size Interstitial Moving Phase Volume... 

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